For \(k \ge 7\) , the best construction comes from a result of Bohman and Keevash [1]} on random graph processes, and is of the form
\(\operatorname{ex}(n,K_{k,k}) = \Omega \left(n^{2 - 2/(k+1)} (\log k)^{1/(k^2-1)}\right).\) 
